i
DELAMINATION EFFECT ON RESPONSE OF A
COMPOSITE BEAM BY WAVELET SPECTRAL
FINITE ELEMENT METHOD
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF
REQUIREMENTS FOR THE DEGREE OF
MASTER OF TECHNOLOGY
IN
CIVIL ENGINEERING
BY
VENNA VENKATESWARAREDDY
210CE2030
Under the guidance of
DR. MANORANJAN BARIK
DEPARTMENT OF CIVIL ENGINEERING
NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA, ORISSA.
2012
ii
Dr. Manoranjan Barik
Associate Professor
Department of Civil Engineering
National Institute of Technology, Rourkela, Orissa, India. ---------------------------------------------------------------------------------------------------------------------
Certificate
This is to certify that the thesis entitled “DELAMINATION EFFECT ON RESPONSE OF A
COMPOSITE BEAM BY WAVELET SPECTRAL FINITE ELEMENT METHOD”
submitted by Mr. VENNA VENKATESWARAREDDY, bearing Roll No. 210CE2030 in
partial fulfilment of the requirements for the award of Master of Technology Degree in Civil
Engineering with specialization in Structural Engineering during 2010-2012 session at the
National Institute of Technology, Rourkela is an authentic work carried out by him under my
supervision.
To the best of my knowledge, the matter embodied in the thesis has not been submitted to any
other University/Institute for the award of any degree or diploma.
Rourkela Dr. Manoranjan Barik
Date: Associate Professor
Department of Civil Engineering
NIT Rourkela, 769008
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Acknowledgement
I am grateful to the Dept. of Civil Engineering, NIT ROURKELA, for giving me the
opportunity to execute this project, which is an integral part of the curriculum in M.Tech
programme at the National Institute of Technology, Rourkela.
I consider it my privilege to express my gratitude, respect and sincere thanks to all those
who guided and supported me in successful accomplishment of this project work. First of all I
would like to express my deepest sense of respect and indebtedness to my guide, Dr.
Manoranjan Barik, for his consistent support, guidance, encouragement, precious pieces of
advice in times, the freedom he provided and above all being nice to me round the year, starting
from the first day of my project work up to the successful completion of the thesis work.
My special thanks to Prof. N. Roy, Head of the Civil Engineering Department, for all the
facilities provided to successfully complete this work. I am also very thankful to all the faculty
members of the department, especially Structural Engineering specialization for their constant
encouragement, invaluable advice, encouragement, inspiration and blessings during the project.
The personal communication and discussion with Dr. Mira Mitra of Indian Institute of
Technology, Bombay is of immense help. Her valuable suggestions and timely co-operation
during the project work is highly acknowledged. The author extends his heartfelt thanks to her.
Especially I am thankful to my family members, all my classmates, friends and
Surendra Naga for their constant source of support, inspiration and valuable advices when in
need.
Venna Venkateswarareddy
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PREFACE
Transform methods are very useful to solve the ordinary and partial differential
equations. Fourier and Laplace transforms are the most commonly used transforms. Wavelet
transforms are most popular with electrical and communication engineers to analyse the signals.
From last few years, Wavelet transforms are in use for structural engineering problems, like
solution of ordinary and partial differential equations. Dynamical problems in structural
engineering fall under two categories, one involving low frequencies (structural dynamics
problems) and the other involving high frequencies (wave propagation problems).
Spectral Finite Element (SFE) method is a transform method to solve the high frequency
excitation problems which are encountered in structural engineering. SFE based on Fourier
transforms has high limitations in handling finite structures and boundary conditions. SFE based
with wavelet transforms is a very good tool to analyse the dynamical problems and eliminate
many limitations.
In this project, a model for embedded de-laminated composite beam is developed using
the wavelet based spectral finite element (WSFE) method for the de-lamination effect on
response using wave propagation analysis. The simulated responses are used as surrogate
experimental results for the inverse problem of detection of damage using wavelet filtering. The
technique used to model a structure that, through width de-lamination subdivides the beam into
base-laminates and sub-laminates along the line of de-lamination. The base-laminates and sub-
laminates are treated as structural waveguides and kinematics are enforced along the connecting
line. These waveguides are modeled as Timoshenko beams with elastic and inertial coupling and
the corresponding spectral elements have three degrees of freedom, namely axial, transverse and
v
shear displacements at each node. The internal spectral elements in the region of de-lamination
are assembled assuming constant cross sectional rotation and equilibrium at the interfaces
between the base-laminates and sub-laminates. Finally, the redundant internal spectral element
nodes are condensed out to form two-noded spectral elements with embedded de-lamination. The
response is being obtained by coding programs in MATLAB.
vi
CONTENTS
CHAPTER PAGE No.
CERTIFICATE………………………………………………………………………………...ii
ACKNOWLEDGEMENT…………………………………………………………………...iii
PREFACE………………………………………………………………………………………iv
CONTENTS…………………………………………………………………………………….vi
LIST OF FIGURES...................................................................................................................ix
LIST OF TABLES....................................................................................................................xii
ABBREVATIONS…………………………………………………………………………….xii
SYMBOLS……………………………………………………………………………………..xv
1 INTRODUCTION…………………………………………………………………...........1
1.1 Introduction …………………………………………………………………………...2
1.2 Solution methods for structural dynamic problems…………………………………...2
1.3 Solution methods for wave propagation problems…………………………………...4
2 LITERATURE REVIEW…………………………………………………………..........7
2.1 Natural frequency-based methods……………………………………………………..8
2.2 Mode shape-based methods………………………………………………………….12
2.3 Curvature/strain mode shape-based methods………………………………………...18
2.4 Other methods based on modal parameters………………………………………….20
vii
3 TRANSFORM METHODS…………………………………………………….............22
3.1 Laplace transforms…………………………………………………………………...23
3.2 Fourier transforms………………………………………………………………........25
3.2.1 Continuous Fourier transforms (CFT)…………………………………........25
3.2.2 Fourier Series (FS)…………………………………………………………..28
3.2.3 Discrete Fourier Transform (DFT)………………………………………….30
3.3 Wavelet transforms…………………………………………………………………...31
3.3.1 Types of wavelets…………………………………………………………...32
3.3.1.1 Morlet's Wavelet………………………………………………….32
3.3.1.2 Mexican Hat wavelet……………………………………………..34
3.3.1.3 The Continuous Wavelet Transform……………………………...34
3.3.1.4 Discrete Wavelet Transform………………………………….......35
3.3.2 Multi-resolution analysis………………………………………………........36
4 WAVELET INTEGRALS………………………………………………………...........39
4.1 Evaluation of wavelet integrals………………………………………………………..40
4.2 Construction of Daubechies compactly supported wavelets………………………....40
4.3 Scaling function and Wavelet function ………………………………......44
4.4 Connection Coefficients ( )………………………………………………………...49
4.5 Moment of scaling functions………………………………………………………….49
4.6 Impact load…………………………………………………………………………...52
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5 WAVELET BASED SPECTRAL FINITE ELEMENT FORMULATION…….53
5.1 Reduction of wave equations to ordinary differential equations…………………......54
5.1.1 Wavelet Extrapolation Technique…………………………………………….57
5.1.2 Calculation of wave numbers and wave amplitudes………………………….60
5.2 Spectral finite element formulation…………………………………………………..62
5.3 Modeling of de-lamination in composite beam………………………………………67
6 NUMERICAL EXAMPLES AND CONCLUSIONS…………………………….72
6.1 Wave responses to the impulse load………………………………………………….74
7 REFERENCES……………………………………………………………………...82-91
ix
LIST OF FIGURES
Figure 3.1: (a) Morlet's wavelet: real part………………………………………………………33
(b) Morlet's wavelet Imaginary part………………………………………………...33
Figure 3.2: Mexican Hat wavelet………………………………………………………………..34
Figure 4.1: (a) Scaling function for N=2………………………………………………………..45
(b) Wavelet function for N=2……………………………………………………...45
Figure 4.2: (a) Scaling function for N=4…………………………………………………….....46
(b) Wavelet function for N=4……………………………………………………...46
Figure 4.3: (a) Scaling function for N=12……………………………………………………...47
(b) Wavelet function for N=12…………………………………………………….47
Figure 4.4: (a) Scaling Function for N=22……………………………………………………..48
(b) Wavelet Function for N=22…………………………………………………....48
Figure 4.5: Discredited impulse time signal…………………………………………………….52
Figure 5.1: Composite beam element with nodal forces and nodal displacements……………..62
Figure 5.2: Graphite-Epoxy [0] 8layered cantilever beam ……………………………………..67
x
Figure 5.3: (a) Cross section at the end…………………………………………………………68
(b) Cross section at the de-lamination……………………………………………...68
Figure 5.4: Modeling of an embedded de-lamination with base and sub laminates……….........68
Figure 5.5: Representation of the base and sub base laminates by spectral elements…………..68
Figure 5.6: Force balance at the interface between base and sub laminate elements…………...69
Figure 6.1: Axial tip velocity of undamaged [ ] composite beam………………………...…..74
Figure 6.2: Transverse tip velocity of undamaged [ ] composite beam……………………….75
Figure 6.3: Axial tip velocities for undamaged and different de-lamination lengths
( =10mm and 20mm) along the centerline of [ ] layered composite beam……......76
Figure 6.4: Transverse tip velocities for undamaged and different de-lamination lengths
( =10mm and 20mm) along the centerline of [ ] layered composite beam……...76
Figure 6.5: Axial tip velocities for undamaged and different de-lamination lengths
( =30mm and 40mm) along the centerline of [ ] layered composite beam………77
Figure 6.6: Transverse tip velocities for undamaged and different de-lamination lengths
( =30mm and 40mm) along the centerline of [ ] layered composite beam………77
Figure 6.7: Axial tip velocities for undamaged and different position (0.25m and 0.35m)
of 20mm de-lamination from free end and along the centerline of [ ]
xi
layered composite beam…………………………………………………………….78
Figure 6.8: Transverse tip velocities for undamaged and different position (0.25m and 0.35m)
of 20mm de-lamination from free end and along the centerline of [ ]
composite beam……………………………………………………………………...78
Figure 6.9: Axial tip velocities for undamaged and different depths above the centerline
(h1=h, h1=h/2 and h1=h/4) of 20mm de-lamination of [ ] layered
composite beam……………………………………………………………………..79
Figure 6.10: Transverse tip velocities for undamaged and different depths above
the centerline (h1=h, h1=h/2 and h1=h/4) of 20mm de-lamination of [ ]
layered composite beam…………………………………………………………….80
Figure 6.11: Axial tip velocities for different orientation ( , , , and )
of 20mm de-lamination along the centerline of composite beam………………….80
Figure 6.12: Transverse tip velocities for different orientation ( , , , and )
of 20mm de-lamination along the centerline of composite beam…………………81
xii
LIST OF TABLES
Table 4.1: Filter Coefficients for N=22…………………………………………………………44
Table4. 2: First order (Ω1) and Second order (Ω
2) connection coefficients for N=22………….50
Table 4.3: Moment of Scaling Functions for N= 22…………………………………………….51
Table 6.1: Material properties of the Graphite- Epoxy composite beam……………………….73
xiii
ABBREVATIONS
ANN Artificial Neural Networks
AWCD Approximate Waveform Capacity Dimension
CFT Continuous Fourier Transform
CWT Continuous Wavelet Transform
DI Damage Index
DOF Degrees of Freedom
DST Discrete Fourier Transform
FD Fractal Dimension
FE Finite Element
FEM Finite Element Method
FFT Fast Fourier transform
FIR Finite Impulse Responce
FS Fourier Series
FSFE Fourier Transform based Spectral Finite Element
GFD Generalized Fractal Dimension
xiv
MDLAC Multiple Damage Location Assurance Criterion
MDOF Multi Degree of freedom
MRA Multiresolution Analysis
MSA Multiscale Analysis
MSC Mode Shape Curvature
NN Neural Network
ODE Ordinary Differential Equations
PDE Partial Differential Equations
PEP Polynomial Evaluation Problem
SCFEM Super Convergent Finite Element Method
SCCM Spectral Centre Correction Method
SDI Single Damage Indicator
SQP Sequential Quadratic Programming
SVD Singular Value Decomposition
SWT Stationary Wavelet Transform
ULS Uniform Load Surface
WSFEM Wavelet based Spectral Finite Element Method
xv
SYMBOLS
a Scaling parameter
Filter coefficients
In-plane laminate maduli coefficients
b Translation parameter
Width of the beam
In-plane/ flexure coupling laminate moduli coefficients
Constant coefficients
Approximation coefficients
d Rectangular pulse width
Detailed coefficients
Flexural laminate stiffness coefficient.
Young’s modulus of fibre along the longitudinal axis
Young’s modulus of fibre across the longitudinal axis
F(s) Function in terms of the Laplace s
F(t) Function in terms of time t
xvi
Nodal force vector of the element at node
Global Force vector
Shear modulus of composite in plane
2h Depth of the beam
Imaginary unit
Inertial constants
k Wave number.
Elemental Dynamic Stiffness Matrix
Global Dynamic Stiffness Matrix
L Length of the beam
Distance between the free end and edge of the de-lamination
De-lamination length
M Vanishing moment
Moment in spatial and temporal dimensions
Transformed moment
Number of sampling points
N Order of Debauchies
xvii
Transformed axial force
Axial force in spatial and temporal dimensions
[ ] Amplitude ratio matrix
and Transformation matrices
Time
Final time
Axial displacement in spatial and temporal dimensions
Nodal displacement vectors of the element at node
Global displacement matrix
Sequence of nested subspaces
Transverse force in spatial and temporal dimensions
Transformed transverse force
Transverse displacement in spatial and temporal dimensions
Thickness of each ply of laminate
pi
Circular frequency
Discrete circular frequency in
xviii
Scaling functions
Wavelet functions
Mother wavelet
First derivative of scaling function
Second derivative of scaling function
Direct sum
Orthogonal to each other
derivative of Connection coefficients
moment of scaling function
Time in micro second
Shear displacement in spatial and temporal dimensions
Mass density
First order connection coefficient matrix
Second order connection coefficient matrix
Eigenvector matrix of
Diagonal matrix containing corresponding eigenvalues
and Diagonal matrices
2
CHAPTER-1
INTRODUCTION
1.1 Introduction
Over the past few decades the composite materials are extensively used in many
engineering fields such as civil, mechanical and aerospace engineering etc. In-plane properties
of the composite material are much higher than its transverse tensile and inter-laminar shear
strength. Due to less strength in transverse direction composite structures are very much prone to
defects like matrix cracking, fibre fracture, fibre de-bonding, de-lamination/inter-laminar de-
bonding, of which de-lamination is most common, easily exposed to damage and it may increase,
thus reducing the life of the structure. Structural components are often subjected to damage
which can potentially reduce the safety. It is very important to find the weakest location and to
detect damage at the earliest possible stage to avoid brittle failure in future. This technique
(WSFEM) is based on the response-based approach since the response data are directly related to
damage. This approach is therefore fast and inexpensive.
1.2 Solution methods for structural dynamic problems
Generally dynamic analysis of the structure can be done by Finite Element Method
(FEM). In structural engineering, dynamic analysis of structures can be divided into two
categories, one is related with the low frequency loading categorized as structural dynamic
problems and another related with high frequency loading categorized as wave propagation
problems. Most of the structures of dynamic analysis come under structural dynamics. In
structural dynamics problems , the solution can be determined either by system parameters such
as natural frequencies and mode shapes or in terms of simulated response of the system to the
3
external excitation such as initial displacements support motion and applied load etc. The first
few mode shapes and natural frequencies are sufficient to analyze the performance of the
structure.
Finite Element solution of the dynamic analysis of the structure can be obtained by two
different methods [5] which are Modal Methods and Time Marching Schemes. In general, for
multi degree of freedom system (MDOF), the governing wave equation is coupled with a set of
ordinary differential equations (ODE). Linear transformation of the above ordinary differential
equations linearly and decoupled by modal matrix are referred as modal analysis. Modal
analysis is also an eigenvalue analysis and it is like one of the several numerical techniques such
as matrix iteration method. Simple continuous systems like rod, beam, plates can be solved
analytically. The solutions of such continuous systems are restricted though they are exact. But
complicated structures can be solved by approximate techniques. In general, approximate
techniques convert the continuous systems into discrete systems. These approximate techniques
are grouped into two categories. In the first group, the solution in terms of known functions are
assumed and they are combined linearly. For the continuous system, the governing differential
equations are partial differential equations. These partial differential equations are converted to a
set of ordinary differential equations by substituting the assumed solution with unknown
coefficient as variables. The Rayleigh-Ritz and Galerkin methods are the examples of this
method. In the other group, the dynamics of the continuous system is shown in terms of large
number of discrete points on the system. The finite element technique falls under this group. In
this finite element method the continuous structure is divided into number of elements and each
element connects through nodes surrounded by it. The continuous system of structure can be
reduced to multi degree of freedom system by expressing the dynamics in terms of the
4
displacements of the nodes. These displacements are approximated by some functions, the
coefficients of these functions are obtained in terms of the displacements of the nodes. This finite
element method can be applied to any arbitrary shapes and structure having high complexities.
Other methods such as boundary element method [7, 39] and meshless methods [6] and wave
finite element method [66]are applied for solving structural dynamic problems.
1.3 Solution methods for wave propagation problems
Multi modal problems are related with wave propagation analysis, in which the
extraction of eigenvalues is computationally most expensive. So Modal Methods are not suitable
for multi modal problems which are having very high frequencies. In the wave propagation
problems, the short term effects are critical, because the frequency of the input loading is very
high. To get the accurate mode shapes and natural frequencies, the wave length and mesh size
should be small. Alternatively we can use the time marching schemes under the finite element
environment. In this method, analysis is performed over a small time step, which is a fraction of
total time for which response histories are required. For some time marching schemes, a
constraint is placed on the time step, and this, coupled with very large mesh sizes, make the
solution of wave propagation problem. Wave propagation deals with loading of very high
frequency content and finite element (FE) formulation for such problems is computationally
prohibitive as it requires large system size to capture all the higher modes. These problems are
usually solved by assuming solution to the field variables say displacements such that the
assumed solution satisfies the governing wave equation as closely as possible. It is very difficult
to assume a solution in time domain to solve the governing wave equation. Therefore, ignore the
inertial part is ignored and the static part of the governing wave equation is solved exactly, and
this solution is used to obtain mass and stiffness matrices. This method of develop of finite
5
element is called Super Convergent Finite Element Method (SCFEM) [20, 9, 10, 42, 11]. This
method gives the smaller system size for wave propagation problems than conventional Finite
Element Method (FEM).
Alternatively, the solution in frequency domain is assumed and the governing equations
solve are transformed and solved exactly. This simplifies the problem by introducing the
frequency as a parameter which removes the time variable from the governing equations by
transforming to the frequency domain. Among these techniques, many methods are based on
integral transforms [22] which include Laplace transform, Fourier transform and most recently
wavelet transform. The solution of these transformed equations is much easier than the original
partial differential equations. The main advantage of this system is computational efficiency over
the finite element solution. These solutions in transformed frequency domain contain information
of several frequency dependent wave properties essential for the analysis. The time domain
solution is then obtained through inverse transform. In the frequency domain Fourier methods
can be used to achieve high accuracy in numerical differentiation. One such method is FFT based
spectral finite element method. The WSFE technique is very similar to the fast Fourier transform
(FFT) based spectral finite element (FSFE) except that it uses compactly supported Daubechies
scaling function approximation in time. In FSFEM, first the governing PDEs are transformed to
ODEs in spatial dimension using FFT in time. These ODEs are then usually solved exactly,
which are used as interpolating functions for FSFE formulation.
The advantages of FSFEM are, they reduces the system size and the wave characteristics
can be extracted directly from such formulation. The main drawback of FSFEM is that it cannot
handle waveguides of short lengths. This is because the required assumption of periodicity
results in wrap around problems, which totally distorts the response. It is in such cases,
6
compactly supported wavelets, which have localized basis functions can be efficiently used for
waveguides of short lengths. The wavelet based spectral finite element method follows an
approach very similar to FSFEM, except that Daubechies scaling functions are used for
approximation in time for reduction of PDEs to ODEs. The approach removes the problem
associated with ‘wrap around’ in FSFEM and thus requires a smaller time window for the same
problem. The Fourier transform is a tool widely used for many scientific purposes, but it is well
suited only to the study of stationary signals where all frequencies have an infinite coherence
time. The Fourier analysis brings only global information which is not sufficient to detect
compact patterns.
8
CHAPTER-2
LITERATURE REVIEW
This review is organized by the classification using the features extracted for damage
identification, and these damage identification methods are categorized as follows:
1. Natural frequency-based methods;
2. Mode shape-based methods;
3. Curvature/strain mode shape-based methods;
4. Other methods based on modal parameters.
2.1 NATURAL FREQUENCY-BASED METHODS:- Natural frequency-based
methods use the natural frequency change as the basic feature for damage identification. The
choice of the natural frequency change is attractive because the natural frequencies can be
conveniently measured from just a few accessible points on the structure and are usually less
contaminated by experimental noise.
Liang et al. [36] developed a method based on three bending natural frequencies for the
detection of crack location and quantification of damage magnitude in a uniform beam under
simply supported or cantilever boundary conditions. The method involves representing crack as a
rotational spring and obtaining plots of its stiffness with crack location for any three natural
modes through the characteristic equation. The point of intersection of the three curves gives the
crack location and stiffness. The crack size is then computed using the standard relation between
stiffness and crack size based on fracture mechanics. This method had been extended to stepped
beams by Nandwana and Maiti [52] and to segmented beams by Chaudhari and Maiti [13]
using the Frobenius method to solve Euler-Bernoulli type differential equations.
9
Chinchalkar [14] used a finite element-based numerical approach to mimic the semi-analytical
approach using the Frobenius method. The beam is modelled using beam elements and the
inverse problem of finding the spring stiffness, given the natural frequency, is shown to be
related to the problem of a rank-one modification of an eigenvalue problem. This approach does
not require quadruple precision computation and is relatively easy to apply to different boundary
conditions. The results are compared with those from semi-analytical approaches. The biggest
advantage of this method is the generality in the approach; different boundary conditions and
variations in the depth of the beam can be easily modelled.
Morassi and Rollo [47] showed that the frequency sensitivity of a cracked beam-type structure
can be explicitly evaluated by using a general perturbation approach. Frequency sensitivity turns
to be proportional to the potential energy stored at the cracked cross section of the undamaged
beam. Moreover, the ratio of the frequency changes of two different modes turns to be a function
of damage location only. Morassi’s method based on Euler–Bernoulli beam theory modeled
crack and as a massless, infinitesimal rotational spring. The explicit expression is valid only for
small defects.
Kasper et al. [30] derived the explicit expressions of wave number shift and frequency shift for
a cracked symmetric uniform beam. These expressions apply to beams with both shallow and
deeper cracks. But the explicit expressions are based on high frequency approximation, and
therefore, they are generally inaccurate for the fundamental mode and for a crack located in a
boundary-near field.
Messina et al. [40] proposed a correlation coefficient termed the multiple damage location
assurance criterion (MDLAC) by introducing two methods of estimating the size of defects in a
structure. The method is based on the sensitivity of the frequency of each mode to damage in
10
each location. ‘MDLAC’ is defined as a statistical correlation between the analytical predictions
of the frequency changes f and the measured frequency changes f. The analytical frequency
change f can be written as a function of the damage extent vector D. The required damage
state is obtained by searching for the damage extent vector D which maximizes the MDLAC
value. Two algorithms (i.e., first and second order methods) were developed to estimate the
absolute damage extent. Both the numerical and experimental test results were presented to show
that the MDLAC approach offers the practical attraction of only requiring measurements of the
changes in a few of natural frequencies of the structure between the undamaged and damaged
states and provides good predictions of both the location and absolute size of damage at one or
more sites.
Lele and Maiti[34] extended Nandwana and Maiti’s method [53] to short beam, taking into
account the effects of shear deformation and rotational inertia through Timoshenko beam theory.
Patil and Maiti proposed a frequency shift-based method for detection of multiple open cracks in
an Euler–Bernoulli beam with varying boundary conditions. This method is based on the transfer
matrix method and extends the scope of the approximate method given by Liang for a single
segment beam to multi-segment beams. Murigendrappa et al. [50] later applied Patil and
Maiti’s approach to single/multiple crack detection in pipes filled with fluid.
Morassi [45] presented a single crack identification in a vibrating rod based on the knowledge
of the damage-induced shifts in a pair of natural frequencies. The analysis is based on an explicit
expression of the frequency sensitivity to damage and enables non uniform bars under general
boundary conditions to be considered. Some of the results are also valid for cracked beams in
bending. Morassi and Rollo [47] later extended the method to the identification of two cracks of
equal severity in a simply supported beam under flexural vibrations.
11
Kim and Stubbs [31] proposed a single damage indicator (SDI) method to locate and quantify a
crack in beam-type structures by using changes in a few natural frequencies. A crack location
model and a crack size model were formulated by relating fractional changes in modal energy to
changes in natural frequencies due to damage. In the crack location model, the measured
fractional change in the ith eigenvalue Zi and the theoretical (FEM based) modal sensitivity of the
ith
modal stiffness with respect to the jth element Fij is defined. The theoretical modal curvature is
obtained from a third order interpolation function of theoretical displacement mode shape. Then,
an error index eij is introduced to represent the localization error for the ith mode and the j
th
location. SDI is defined to indicate the damage location. While in the crack size model, the
damage inflicted aj at predefined locations can be predicted using the sensitivity equation. The
crack depth can be computed from aj and the crack size model based on fracture mechanics. The
feasibility and practicality of the crack detection scheme were evaluated by applying the
approach to the 16 test beams.
Zhong et al. [73] recently proposed a new approach based on auxiliary mass spatial probing
using the spectral centre correction method (SCCM), to provide a simple solution for damage
detection by just using the output-only time history of beam-like structures. A SCCM corrected
highly accurate natural frequency versus auxiliary mass location curve is plotted along with the
curves of its derivatives (up to third order) to detect the crack. However, only the FE verification
was provided to illustrate the method. Since it is not so easy to get a high resolution natural
frequency versus auxiliary mass location curve in experiment as in numerical simulation, the
applicability and practicality of the method in in-situ testing or even laboratory testing are still in
question.
12
Kim, B.H et al. [32] presented a vibration-based damage monitoring scheme to give warning of
the occurrence, location, and severity of damage under temperature induced uncertainty
conditions. A damage warning model is selected to statistically identify the occurrence of
damage by recognizing the patterns of damage driven changes in natural frequencies of the
structure and by distinguishing temperature-induced off-limits.
Jiang et al. [29] incorporated a tunable piezoelectric transducer circuitry into the structure to
enrich the modal frequency measurements, meanwhile implementing a high-order identification
algorithm to sufficiently utilize the enriched information. It is shown that the modal frequencies
can be greatly enriched by inductance tuning, which, together with the high-order identification
algorithm, leads to a fundamentally-improved performance on the identification of single and
multiple damages with the usage of only lower-order frequency measurements.
In 1997 Salawu [64] presented an extensive review of publications before 1997 dealing with the
detection of structural damage through frequency changes. In the conclusion of this review
paper, Salawu suggested that natural frequency changes alone may not be sufficient for a unique
identification of the location of structural damage because cracks associated with similar crack
lengths but at two different locations may cause the same amount of frequency change.
2.2 MODE SHAPE-BASED METHODS:-
Compared to using natural frequencies, the advantage of using mode shapes and their derivatives
as a basic feature for damage detection is quite obvious. First, mode shapes contain local
information, which makes them more sensitive to local damages and enables them to be used
directly in multiple damage detection. Second, the mode shapes are less sensitive to
environmental effects, such as temperature, than natural frequencies. The disadvantages are also
13
apparent. First, measurement of the mode shapes requires a series of sensors; second, the
measured mode shapes are more prone to noise contamination than natural frequencies.
Shi et al. [65] extended the damage localization method based on multiple damage location
assurance criterions (MDLAC) by using incomplete mode shape instead of modal frequency.
The two-step damage detection procedure is to preliminarily localize the damage sites by using
incomplete measured mode shapes and then to detect the damage site and its extent again by
using measured natural frequencies. No expansion of the incomplete measured mode shapes or
reduction of finite element model is required to match the finite-element model, and the
measured information can be used directly to localize damage sites. The method was
demonstrated in a simulated 2D planar truss model. Comparison showed that the proposed
method is more accurate and robust in damage localization with or without noise effect than the
original MDLAC method. In this method, the use of mode shape is only for preliminary damage
localization, and the accurate localization and quantification of damage still rely on measured
frequency changes.
Lee et al. [33] presented a neural network based technique for element-level damage
assessments of structures using the mode shape differences or ratios of intact and damaged
structures. The effectiveness and applicability of the proposed method using the mode shape
differences or ratios were demonstrated by two numerical example analyses on a simple beam
and a multi-girder bridge.
Hu and Afzal [28] proposed a statistical algorithm for damage detection in timber beam
structures using difference of the mode shapes before and after damage. The different severities
of damage, damage locations, and damage counts were simulated by removing mass from intact
14
beams to verify the algorithm. The results showed that the algorithm is reliable for the detection
of local damage under different severities, locations, and counts.
Pawar et al. [54] investigated the effect of damage on beams with clamped boundary conditions
using Fourier analysis of mode shapes in the spatial domain. The damaged mode shapes were
expanded using a spatial Fourier series, and a damage index (DI) in the form of a vector of
Fourier coefficients was formulated. A neural network (NN) was trained to detect the damage
location and size using Fourier coefficients as input. Numerical studies showed that damage
detection using Fourier coefficients and neural networks has the capability to detect the location
and damage size accurately. However, the use of this method is limited to beams with clamped-
clamped boundary condition.
Abdo and Hori [2] suggested that the rotation (i.e., the first derivative of displacement) of mode
shape is a sensitive indicator of damage. Based on a finite element analysis of a damaged
cantilevered plate and a damaged simply-supported plate, the rotation of mode shape is shown to
have better performance of multiple damage localization than the displacement mode shape
itself.
Hadjileontiadis et al. [24] and Hadjileontiadis and Douka [25] proposed a response-based
damage detection algorithm for beams and plates using Fractal Dimension (FD). This method
calculates the localized FD of the fundamental mode shape directly. The damage features are
established by employing a sliding window of length M across the mode shape and estimating
the FD at each position for the regional mode shape inside the window. Damage location and
size are determined by a peak on the FD curve indicating the local irregularity of the
fundamental mode shape introduced by the damage. If the higher mode shapes were considered,
this method might give misleading information as demonstrated in their study.
15
Wang and Qiao [69] proposed a modified FD method termed ‘generalized fractal dimension
(GFD)’ method by introducing a scale factor S in the FD algorithm, Instead of directly applying
the algorithm to the fundamental mode shape, the GFD is applied to the ‘uniform load surface’
(ULS) to detect the damage. Three different types of damage in laminated composite beams have
been successfully detected by the GFD. It should be pointed out that the GFD bears no
conventional physical meaning as compared to the FD, and it only serves as an indicator of
damage. A scale factor S has to be carefully chosen in order to detect damage successfully.
Qiao and Cao [58] proposed a novel waveform fractal dimension-based damage identification
algorithm. An approximate waveform capacity dimension (AWCD) was formulated first, from
which an AWCD-based modal irregularity algorithm (AWCD-MAA) was systematically
established. Then, the basic characteristics of AWCD-MAA on irregularity detection of mode
shapes, e.g., crack localization, crack quantification, noise immunity, etc., were investigated
based on an analytical crack model of cantilever beams using linear elastic fracture mechanics.
In particular, from the perspective of isomorphism, a mathematical solution on the use of
applying waveform FD to higher mode shapes for crack identification was originally proposed,
from which the inherent deficiency of waveform FD to identify crack when implemented to
higher mode shapes is overcome. The applicability and effectiveness of the AWCD-MAA was
validated by an experimental program on damage identification of a cracked composite
cantilever beam using directly measured strain mode shape from smart piezoelectric sensors.
Hong et al. [27] showed that the continuous wavelet transform (CWT) of mode shape using a
Mexican hat wavelet is effective to estimate the Lipschitz exponent for damage detection of a
damaged beam. The magnitude of the Lipschitz exponent can be used as a useful indicator of the
16
damage extent. It was also proved in their work that the number of the vanishing moments of
wavelet should be at least 2 for crack detection in beams.
Douka et al. [17, 18] applied 1D symmetrical 4 wavelet transform on mode shape for crack
identification in beam and plate structures. The position of the crack is determined by the sudden
change in the wavelet coefficients. An intensity factor is also defined to estimate the depth of the
crack from the coefficients of the wavelet transform.
Zhong and Oyadiji [72] proposed a crack detection algorithm in symmetric beam-like structures
based on stationary wavelet transform (SWT) of mode shape data. Two sets of mode shape data,
which constitute two new signal series, are, respectively, obtained from the left half and
reconstructed right half of modal displacement data of a damaged simply supported beam. The
difference of the detail coefficients of the two new signal series was used for damage detection.
The method was verified using modal shape data generated by a finite element analysis of 36
damage cases of a simply supported beam with an artificial random noise of 5% SNR. The
effects of crack size, depth and location as well as the effects of sampling interval were
examined. The results show that all the cases can provide evidence of crack existence at the
correct location of the beam and that the proposed method can be recommended for
identification of small cracks as small as 4% crack ratio in real applications with measurement
noise present. However, there are two main disadvantages of this method. First, the use of this
method based on SWT requires fairly accurate estimates of the mode shapes. Second, the method
cannot tell the crack location from its mirror image location due to its inherent limitation.
Therefore, in applying the method, both the crack location predicted and its mirror image
location should be checked for the presence of a crack.
17
Chang and Chen [12] presented a spatial Gabor wavelet-based technique for damage detection
of a multiple cracked beam. Given natural frequencies and crack positions, the depths of the
cracks are then solved by an optimization process based on traditional characteristic equation.
Analysis and comparison showed that it can detect the cracks positions and depths and also has
high sensitivity to the crack depth, and the accuracy of this method is good. The limitation of this
method is very common in wavelet transform methods, that is, there are peaks near the
boundaries in the wavelet plot caused by discontinuity and the crack cannot be detected when the
crack is near the boundaries.
Cao and Qiao [15] proposed a novel wavelet transform technique (so called ‘integrated wavelet
transform’), which takes synergistic advantage of the SWT and the CWT, to improve the
robustness of irregularity analysis of mode shapes in damage detection. Two progressive wavelet
analysis steps are considered, in which the SWT-based multiresolution analysis (MRA) is first
employed to refine the retrieved mode shapes, followed by CWT-based multiscale analysis
(MSA) to magnify the effect of slight irregularity. The SWT-MRA is utilized to separate the
multi-component modal signal, eliminate random noise and regular interferences, and thus
extract purer damage information; while the CWT-MSA is employed to smoothen, differentiate
or suppress polynomials of mode shapes to magnify the effect of irregularity. The choice of the
optimal mother wavelet in damage detection is also elaborately discussed. The proposed
methodology is evaluated using the mode shape data from the numerical finite element analysis
and experimental testing of a cantilever beam with a through-width crack. The methodology
presented provides a robust and viable technique to identify minor damage in a relatively lower
signal-to-noise ratio environment.
18
2.3 CURVATURE/STRAIN MODE SHAPE-BASED METHODS:- It has been
shown by many researchers that the displacement mode shape itself is not very sensitive to small
damage, even with high density mode shape measurement. As an effort to enhance the sensitivity
of mode shape data to the damage, the mode shape curvature (MSC) is investigated as a
promising feature for damage identification.
Pandey et al. [53] suggested for the first time that the MSC, that is, the second derivatives of
mode shape, are highly sensitive to damage and can be used to localize it. The curvature mode
shapes are derived using a central difference approximation. Result showed that the difference of
curvature mode shapes from intact and damaged structure can be a good indicator of damage
location. It is also pointed out that for the higher modes, the difference in modal curvature shows
several peaks not only at the damage location but also at other positions, which may lead to a
false indication of damage. Hence, in order to reduce the possibility of a false alarm, only first
few low curvature mode shapes can be used for damage identification.
Abdel Wahab and De Roeck [1] investigated the accuracy of using the central difference
approximation to compute the MSC based on finite element analysis. The authors suggested that
a fine mesh is required to derive the modal curvature correctly for the higher modes and that the
first mode will provide the most reliable curvature in practical application due to the limited
number of sensors needed. Then, a damage indicator called ‘curvature damage factor’, which is
the average absolute difference in intact and damaged curvature mode shapes of all modes, is
introduced. The technique is further applied to a real structure, namely bridge Z24, to show its
effectiveness in multiple damage location.
Swamidas and Chen [68] performed a finite element-based modal analysis on a cantilever plate
with a small crack. It was found that the surface crack in the structure will affect most of the
19
modal parameters, such as the natural frequencies of the structure, amplitudes of the response
and mode shapes. Some of the most sensitive parameters are the difference of the strain mode
shapes and the local strain frequency response functions. By monitoring the changes in the local
strain frequency response functions and the difference between the strain mode shapes, the
location and severity of the crack that occurs in the structure can be determined.
Li et al. [35] presented a crack damage detection using a combination of natural frequencies and
strain mode shapes as input in artificial neural networks (ANN) for location and severity
prediction of crack damage in beam-like structures. In the experiment, several steel beams with
six distributed surface-bonded strain gauges and an accelerometer mounted at the tip were used
to obtain modal parameters such as resonant frequencies and strain mode shapes.
Amaravadi et al [4] proposed an orthogonal wavelet transform technique that operates on
curvature mode shape for enhancing the sensitivity and accuracy in damage location. First, the
curvature mode shape is calculated by central difference approximation from the displacement
mode shapes experimentally obtained from SLV. Then, a threshold wavelet map is constructed
for the curvature mode shape to detect the damage. The experimental results are reasonably
accurate.
Kim et al. [32] proposed a curvature mode shape-based damage identification method for beam-
like structures using wavelet transform. Using a small damage assumption and the Haar wavelet
transformation, a set of linear algebraic equations is given by damage mechanics. With the aid of
singular value decomposition, the singularities in the damage mechanism were discarded.
Finally, the desired DI was reconstructed using the pseudo-inverse solution. The performance of
the proposed method was compared with two existing NDE methods for an axially loaded beam
without any special knowledge about mass density and an applied axial force. The effect of
20
random noise on the performance was examined. The proposed method was verified by a finite
element model of a clamped-pinned pre-stressed concrete beam and by field test data on the I-40
Bridge over the RioGrande
Shi et al. [65] presented a damage localization method for beam, truss or frame type structures
based on the modal strain energy change. The MSEC at the element level is suggested as an
indicator for damage localization.
2.4 OTHER METHODS BASED ON MODAL PARAMETERS:-
Ren and De Roeck [60] proposed a damage identification technique from the finite element
model using frequencies and mode shape change. The element damage equations have been
established through the eigenvalue equations that characterize the dynamic behavior. Several
solution techniques are discussed and compared. The results show that the SVD-R method based
on the singular value decomposition (SVD) is most effective. The method has been verified by a
simple beam and a continuous beam numerical model with numbers of simulated damage
scenarios. The method is further verified by a laboratory experiment of a reinforced concrete
beam.
Wang and Qiao [69] developed a general order perturbation method involving multiple
perturbation parameters for eigenvalue problems with changes in the stiffness parameters. The
perturbation method is then used iteratively with an optimization method to identify the stiffness
parameters of structures. The generalized inverse method is used efficiently with the first order
perturbations, and the gradient and quasi-Newton methods are used with the higher order
perturbations.
Rahai et al. [59] presents a finite element-based approach for damage detection in structures
utilizing incomplete measured mode shapes and natural frequencies. Mode shapes of a structure
21
are characterized as a function of structural stiffness parameters. More equations were obtained
using elemental damage equation which requires complete mode shapes. This drawback is
resolved by presenting the mode shape equations and dividing the structural DOFs to measured
and unmeasured parts. The nonlinear optimization problem is then solved by the sequential
quadratic programming (SQP) algorithm. Monte Carlo simulation is applied to study the
sensitivity of this method to noise in measured modal displacements.
Hao and Xia [26] applied a genetic algorithm with real number encoding to minimize the
objective function, in which three criteria are considered: the frequency changes, the mode shape
changes, and a combination of the two. A laboratory tested cantilever beam and a frame structure
were used to verify the proposed technique. The algorithm did not require an accurate analytical
model and gave better damage detection results for the beam than the conventional optimization
method.
Ruotolo and Surace [63] utilized genetic algorithm to solve the optimization problem. The
objective function is formulated by introducing terms related to global damage and the dynamic
behavior of the structure, i.e., natural frequencies, mode shapes and modal curvature. The
damage assessment technique has been applied to both the simulated and experimental data
related to cantilevered steel beams, each one with a different damage scenario. It is demonstrated
that this method can detect the presence of damage and estimate both the crack positions and
sizes with satisfactory precision. The problems related to the tuning of the genetic search and to
the virgin state calibration of the model are also discussed.
23
CHAPTER-3
TRANSFORM METHODS
3.1 LAPLACE TRANSFORMS
Laplace transforms provide a method for representing and analyzing linear systems using
algebraic methods. The application of Laplace Transform methods is particularly effective for
linear ODEs with constant coefficients, and for systems of such ODEs. To transform an ODE,
we need the appropriate initial values of the function involved and initial values of its
derivatives. In systems that begin undeflected and at rest the Laplace‘s’ can directly replace the
d/dt operator in differential equations [70]. It is a superset of the phasor representation in that it
has both a complex part, for the steady state response, but also a real part, representing the
transient part. As with the other representations the Laplace “s” is related to the rate of change in
the system.
where,
F(s)= the function in terms of the Laplace s
F(t)= the function in terms of time t
The normal convention is to show the function of time with a lower case letter, while the same
function in the s-domain is shown in upper case. Another useful observation is that the transform
starts at s.The Laplace method is particularly advantageous for input terms that are
piecewise defined, periodic or impulsive.
24
Properties of Laplace transform:
1. Linearity:
2. First derivative:
3. Second derivative:
4. Higher order derivative:
5.
where this also implies
6.
where this implies
The original time signal can be obtained through an inverse transform of and is
written as
if
Let us consider for the Laplace transform is
This transform can be performed only analytically and there is no numerical implementation.
This restricts the use of Laplace transform for analysis of problems with higher complexities,
which need to be solved numerically.
25
3.2 FOURIER TRANSFORMS
The Fourier transform is a tool, this can adopt easily and spread throughout system and it
is used in many fields of science as a mathematical or physical tool to alter a problem into one
that can be more easily solved. The Fourier transform, in essence, decomposes or separates a
waveform or function into sinusoids of different frequency which sum to the original waveform.
It identifies or distinguishes the different frequency sinusoids and their respective amplitudes.
The main advantage of Fourier transform in structural dynamics and wave propagation problems
is that several important characteristics of system can be obtained directly from the transformed
frequency domain method. Fourier transform can be implemented analytically, semi analytically
and numerically in the form of Continuous Fourier Transform (CFT), Fourier Series (FS) and
Discrete Fourier Transform (DST) respectively.
In Fourier analysis the complex exponential function is often used. We have by the
Euler formula
Hence, can be considered as complex sinusoid. In general a complex valued function has the
form of
here is the real part of the and is the imaginary part of the , and both
are real valued functions.
3.2.1 CONTINUOUS FOURIER TRANSFORMS
The both forward and inverse continuous Fourier transform of any time signal , generally
shown as transform pair, which are
26
where is the CFT of , is the angular frequency, is the complex number ( ).
should be complex and a plot of amplitude of this function against frequency will give the
spectral density of the time signal . As an example, consider a rectangular pulse width‘d’.
this function can be written as
Substitute the equation (3.12) in the equation (3.11) of , we get
Now, allow the pulse to propagate in the time domain by an amount of seconds, and this can
be written in mathematical form
Substitute the above equation (3.14) in equation (3.13), we get
The magnitude of the equations (3.13) and (3.15) are same. The second transform exists phase
information. The change of phase in frequency domain refers the propagation of signal in the
time domain. By using CFT, spread of the signal can be calculated in the time and frequency
domain.
27
Properties of CFT
(a) Linearity: Consider, two functions for the incident and reflected waves
respectively in time and are the constants. The Fourier Transform of a function
can be obtained as This expression can also be
written as The symbol represents the Fourier
Transform.
(b) Scaling: If multiply with some non zero constant to the time signal will become
The Continuous Fourier Transform of the signal can be written as
This represents that compression occurs in time domain results and
dilation occurs in frequency domain results. The amplitude however decreases to keep
the energy constant.
(c) Time shifting: A shift in the time signal by is manifested as a phase change in the
transformed frequency domain obtained through Continuous Fourier Transform. After
shifting the time the transform pair can be written as .
(d) Symmetric property of the CFT: The CFT of the time signal is in complex form,
split this complex form into real and imaginary parts by using equation (3.9). The real
and imaginary part of the CFT can be written as
The first integral is an even function and second is an odd function.
That is .
28
Consider a point on CFT say origin ( ), the transform on the right side of the origin
can be written as . Similarly on the left side can be written as
This origin point is called
Nyquist frequency and it is very important to determine the half of the total frequency
rang.
(e) Convolution: This property of the CFT is very useful for understanding the signal
processing aspects and it has great importance in wave propagation analysis. This
property can be obtained by multiplying two time signals each other.
Substitute the equation (3.11) in equation (3.18) for both these functions written as
The above equation (3.19) can also be written as
Conversely we can also write as
3.2.2 FOURIER SERIES
Fourier series is in between the Continuous Fourier Transform (CFT) and Discrete
Fourier Transform (DFT). The inverse transform is represented by a series while forward
29
transform is still in the integral form as in the CFT. i.e. out of both equations, one still needs the
mathematical description of the time signal to get the transforms.
The Fourier Series (FS) can be written as
where
The equation (3.22) and equation (3.23) are corresponding to the inverse transform and forward
transform of the CFT respectively. Where is the discrete representation of time signal, and it
introduces the periodicity of time signal. The equations (3.22) and (3.23) can be writing in terms
of complex exponentials as
where
, and the signal repeats after seconds due to enforced periodicity.
Now this time signal, can also be write in terms of fundamental frequency as
where the fundamental frequency
in radians per second or Hz.
Consider the rectangular time signal and substitute in equation (3.25), we get
30
3.2.3 DISCRETE FOURIER TRANSFORM
The transform pair of CFT requires mathematical description of the time signal and their
integration. Data of the time signals are obtained from experiments. Hence, what we require is
the numerical representation for the transform pair, which is called the Discrete Fourier
Transform (DFT). It is an alternative way of mathematical representation of CFT in terms of
summations. Here the ultimate aim is to replace the integral involved in computation of the
Fourier coefficient by summation for numerical implementation. Divide the time signal into
number of constant triangles. The height of the triangle is and width of the triangle is
.
From equation (3.15) we can observe that the continuous transform of the rectangle is a
function. Similarly in discrete system, idealizing the signal as rectangle. Therefore, the DFT of
the signal will be the summation of functions. Hence the second integral of equation
(3.25) can be written as
If is very small, the value of above equation (3.28) nearly equal to unity. Hence the forward
and inverse Discrete Fourier Transform (DFT) can be written as
31
Here, varies from
The DFT of a real function is symmetric about nyquist frequency as like in the case of CFT. One
side of DFT coefficients about nyquist frequency is the complex conjugate of the coefficients on
other side of it. Thus real points are transformed to
complex points. Nyquist frequency can
be calculated by using the following expression
3.3 WAVELET TRANSFORMS
The word wavelet has been derived from the French word. The equivalent French word ondelette
meaning "small wave" was used by Morlet and Grossmann [23, 48, 49] in the early 1980s. A
wavelet is a wave-like oscillation with an amplitude that starts out at zero, increases, and then
decreases back to zero. It can typically be visualized as a "brief oscillation" like one might see
recorded by a seismograph or heart monitor. Generally, wavelets are purposefully crafted to have
specific properties that make them useful for signal processing. Wavelets can be combined, using
a "shift, multiply and sum" technique called convolution, with portions of an unknown signal to
extract information from the unknown signal. Wavelets are defined by the wavelet
function ψ(t) (i.e. the mother wavelet) and scaling function φ(t) (also called father wavelet) in the
time domain. The wavelet function is in effect a band-pass filter and scaling it for each level
halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an
infinite number of levels would be required. The scaling function filters the lowest level of the
transform and ensures all the spectrum is covered.
Since from the last few decades many researchers showing their interest for the development and
application wavelets in many fields. Some of the researchers include Morlet and Grossmann [23]
32
developed formulation for the Continuous Wavelet Transform, Meyer [41] and Mallt [38]
developed multi resolution analysis using wavelets, Daubechies [16] for proposal of orthogonal
compactly supported wavelets and Stromberg [67] for early works on discrete wavelet transform.
As a mathematical tool, wavelets can be used to extract information from many different kinds of
data of signals and images. Sets of wavelets are generally needed to analyze data fully. A set of
"complementary" wavelets will deconstruct data without gaps or overlap so that the
deconstruction process is mathematically reversible. Thus, sets of complementary wavelets are
useful in wavelet based compression/decompression algorithms where it is desirable to recover
the original information with minimal loss.
3.3.1 TYPES OF WAVELETS
We have several wavelet functions such as Morlet wavelets, Mexican hat wavelets, Mayer
wavelets, Shanon wavelets, Daubechies wavelets e.t.c., out of these wavelet functions Morlet
wavelets and Mexican hat wavelets are explained here, Daubechies wavelet function explained
in chapter 4.
3.3.1.1 MORLET'S WAVELET
The wavelet defined by Morlet is:
(3.32)
It is a complex wavelet which can be decomposed in two parts, one for the real part, and the
other for the imaginary part.
33
where is a constant. The admissibility condition is verified only if . Figure 1shows
the real part of the Morlet wavelet function and figure 2 shows the imaginary part of the Morlet
wavelet function.
Figure 3.1: (a) Morlet's wavelet: real part
Figure 3.1: (b) Morlet's wavelet Imaginary part
-8 -6 -4 -2 0 2 4 6 8-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5Complex Morlet wavelet cmor1.5-1
Real part
-8 -6 -4 -2 0 2 4 6 8-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Imaginary part
34
3.3.1.2 MEXICAN HAT WAVELET
The Mexican hat defined by Murenzi is:
It is the second derivative of a Gaussian (see figure 2).
Figure 3.2: Mexican Hat wavelet.
3.3.1.3 THE CONTINUOUS WAVELET TRANSFORM
The Morlet-Grossmann definition of the continuous wavelet transform for a 1D signal
is:
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Mexican hat wavelet
35
where is the mother wavelet, a (>0) is the scaling parameter, it represents the frequency
content of the wavelet and b is the translation parameter, it represents the location of wavelet in
time. The basis function of the wavelet transform retains the time locality and frequency locality.
The transform is characterized by the following three properties:
1. It is a linear transformation,
2. It is covariant under translations:
3. It is covariant under dilations:
The last property makes the wavelet transform very suitable for analyzing hierarchical structures.
It is like a mathematical microscope with properties that do not depend on the magnification.
3.3.1.4 DISCRETE WAVELET TRANSFORM
The discrete wavelet transform (DWT) can be derived from this theorem if we process a
signal which has a cut-off frequency. A digital analysis is provided by the discretisation of
equation (3.34) with some simple considerations on the modification of the wavelet pattern by
dilation. Usually the wavelet function has no cut-off frequency and it is necessary to
suppress the values outside the frequency band in order to avoid aliasing effects. We can work in
Fourier space, computing the transform scale by scale. The number of elements for a scale can be
reduced, if the frequency bandwidth is also reduced. This is possible only for a wavelet which
also has a cut-off frequency. The decomposition proposed by Littlewood and Paley provides a
very nice illustration of the reduction of elements scale by scale. This decomposition is based on
an iterative dichotomy of the frequency band. The associated wavelet is well localized in Fourier
space where it allows a reasonable analysis to be made although not in the original space. The
36
search for a discrete transform which is well localized in both spaces leads to multiresolution
analysis.
3.3.2 MULTI-RESOLUTION ANALYSIS
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for
each bounded rectangular region in the upper half plane. Still, each coefficient requires the
evaluation of an integral. To avoid this numerical complexity, one needs one auxiliary function,
the father wavelet . The sequence of nested subspaces plays an important role in the
construction of wavelet functions. These nested subspaces represent the multi resolution of
wavelet function in The closed subspaces for with the following properties,
1.
2.
3.
4. if and only if
These properties of subspaces are related with scaling relation. Thus, the embedded
subspaces essentially reduces the problem of obtaining
5. Each subspace is spanned by integers translates of a single function,
Based on the above properties multi resolution analysis, we can conclude that, first we need to
find the scaling function . Such that its integer translates are the
Riesz bases for the space . Similarly, will form a basis for the space . Thus,
37
From equation (3.37) and (3.38), we can conclude that . Therefore the basis function of
space can be write in terms of basis function as
where are filter coefficients. The equation (3.39) gives the scaling function for
the space . The basis function can be defined as
where are the dilation and translation indices respectively. In the above equation
represents the frequency content and represents the time in the analysis of time signal.
Let us consider the approximation of the function into the subspace by the scaling
function as
In the above equation (3.41), are the approximation coefficients and
Similarly the wavelet function and its translates are the Riesz basis for the
subspace , can be expressed in terms of basis functions for as
Where and form the Riesz basis for subspace can be written as
Let us consider the approximation of the function into the subspace by the wavelet
function as
38
Here, are called as detailed coefficients.
The next step is to obtain the closure subspace for the subspace and its orthogonal
compliment such that
where the symbol represents the direct sum. The above equation (3.45) shows that the
subspaces are orthogonal to each other and we can write expression for .
Using the above procedure, calculate the approximations of the at higher scale from the
approximations of lower scale.
40
CHAPTER-4
WAVELET INTEGRALS
4.1 EVALUATION OF WAVELET INTEGRALS
Many wavelet algorithms require the evaluation of integrals which involve combination of
wavelets, scaling functions and their derivatives. We collectively refer to such integrals as
wavelet integrals. There are a few instances in which wavelet integrals can be evaluated
analytically. The accuracy and efficiency of the technique used to compute the integrals will
have a significant impact on the wavelet performance of the wavelet algorithm. For instance
wavelet-based algorithms for solving differential equations typically lead to integrals involving
wavelets and their derivatives. Wavelet integrals[8] which are commonly encountered in
construction of Daubechies compactly supported wavelet algorithms:
(a) Filter coefficients,
(b) Wavelet and Scaling function coefficients,
(c) Moments of Wavelet and Scaling functions and
(d) Connection coefficients.
4.2 CONSTRUCTION OF DAUBECHIES COMPACTLY
SUPPORTED WAVELETS
For a wavelet with compact support, φ(t) can be considered finite in length and is equivalent to
the scaling filter g. An orthogonal wavelet is entirely defined by the scaling filter - a low-pass
41
finite impulse responce (FIR) filter of length 2N and sum 1. In biorthogonal wavelets, separate
decomposition and reconstruction filters are defined. For analysis with orthogonal wavelets the
high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction
filters are the time reverse of the decomposition filters. Daubechies and Symlet wavelets can be
defined by the filter coefficients.
Conditions to calculate the filter coefficients:
1. For uniqueness, normalization is done by considering the area under the scaling function
to be unity,
The above equation leads to the following condition on the filter coefficients,
2. For Daubechies wavelets, the integer translates of scaling functions are orthogonal, i.e.,
where
This gives the condition on the filer coefficients as
42
The conditions given by equations (4.2) and (4.4) are not sufficient to get unique set of filter
coefficients. For an coefficient system,
equations can get from equations (4.2) and
(4.4), remaining
equations can get by imposing some conditions on the wavelet functions.
For Daubechies wavelets, assumed that the scaling functions represents exactly the polynomials
order . Where
Therefore the polynomial order as
The above polynomial is in expanded form, similar to the equation (3.42) for and can be
written as
Since are orthogonal to the translates of , taking inner product of equation (4.6) with
gives
Substitute the equation (4.5) in equation (4.7), we get
The above expression is valid for all values of . Where Consider
and all other gives
43
The above equation can be written in term of filter coefficients, we get
where M is the moment of wavelet function. The scaling functions are obtained by solving
recursively the dilation equation (4.2), (4.4) and (4.5). Which can be expanded for DN as,
Above equation can be written as the following equations,
(0) = a0 (0)
(1) = a0 (2) + a1 (1) + a2 (0)
(2) = a0 (4) + a1 (3) + a2 (2) + a3 (1) + a4 (0)
… … …
(N-2) = aN-3 (N-1) + aN-2 (N-2) + aN-1 (N-3)
(N-1) = aN-1 (N-1)
This can also be written in the matrix form,
The above equation possesses eigenvalue problem and can be solved to obtain as the
eigenvector. The matrix A is known as the filter coefficients and can be solved from
equations. Filter coefficients are given table for N=22. Where N is the order of Daubechies
wavelet. The function in the MATLAB wavelet toolbox gives the filter coefficients.
Table 1 shows the values of filter coefficients for N=22.
44
4.3 SCALING FUNCTION AND WAVELET FUNCTION
Scaling functions and wavelet functions can be calculate by using the following
equations,
where
The following figures are shows the scaling functions and wavelet functions for the
By observing these figures we can say that smoothness increases by
increasing order of Debauchies.
Table 4.1: Filter Coefficients for N=22
Filter Coefficients for N=22
0.0264377294333137 0.0443145095659574
0.203741535201907 0.0294734895982882
0.636254348460787 -0.0217291381089843
0.969707536626371 - 0.00472468792819149
0.582605597780612 0.00696983509023781
-0.229491852355296 -0.000436416206187889
-0.387820982791004 -0.00126292559260676
0.0933997381355307 0.000352354877907889
0.211866179836356 0.0000769884777628895
-0.0657325829045156 -0.0000489812643698126
-0.0939586317975106 6.35586363589251E-06
45
Figure 4.1(a): Scaling function for N=2.
Figure 4.1(b): Wavelet function for N=2.
-1 -0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4scaling function for N=2
-1 -0.5 0 0.5 1-1.5
-1
-0.5
0
0.5
1
1.5wavelet functions for N=2
46
Figure 4.2(a): Scaling function for N=4.
Figure 4.2(b): Wavelet function for N=4.
0 0.5 1 1.5 2 2.5 3-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4scaling function for N=4
0 0.5 1 1.5 2 2.5 3-1.5
-1
-0.5
0
0.5
1
1.5
2wavelet functions for N=4
47
Figure 4.3(a): Scaling function for N=12.
Figure 4.3(b): Wavelet function for N=12.
0 2 4 6 8 10 12-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2scaling function for N=12
0 2 4 6 8 10 12-1.5
-1
-0.5
0
0.5
1
1.5wavelet functions for N=12
48
Figure 4.4(a): Scaling Function for N=22
Figure 4.4(b): Wavelet Function for N=22
0 5 10 15 20 25-0.5
0
0.5
1scaling function for N=22
0 5 10 15 20 25-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1wavelet functions for N=22
49
4.4 CONNECTION COEFFICIENTS
Connection coefficients ( ) are integrals involving combination of wavelets and scaling
functions, their translates and their derivatives. They are frequently encountered during the
discretization of ordinary and partial differential equations. The classes of wavelets for which
connection coefficients are usually desired are those wavelets which are orthogonal, biorthogonal
and compact support. The evaluation of connection coefficients for orthogonal wavelets:
4.5 MOMENT OF SCALING FUNCTIONS
Let be the moment of scaling function
The moment of scaling functions are easily calculated by using the following three recursive
equations given in Amaratunga and Williams (1997) [3],
Always the zero-th moment, of is 1by the normalization of ,
50
Table4. 2: First order (Ω1) and Second order (Ω
2) connection coefficients for N=22.
Connection Coefficients for N=22
0
-3.47337840801057
-0.913209538941252
2.16026933978281
0.347183551085084
-0.602653205128255
-0.145808621125452
0.259980735100996
0.0565725329081428
-0.113990617900921
-0.0189618931390823
0.0444431406647273
0.00525147114923078
-0.0144806539860068
-0.00115026373575154
0.00377470685481452
0.000188643492331173
-0.000752954288200255
-0.0000214714239949848
0.000108557139788138
1.56173825099191E-06
-0.0000103988485588956
-8.57426004680047E-08
5.55791376315321E-07
5.88963918586183E-09
3.72688117300783E-09
3.91513557517705E-10
-5.40977631679387E-09
-1.13407211122615E-10
4.73721978304495E-10
-3.25544726094433E-12
3.16562432692258E-11
-1.60928747640703E-14
2.31293236538858E-13
-1.06221294110902E-17
-1.81854918345231E-15
2.4074144309226E-18
-1.72101206062974E-16
1.95845511016976E-17
1.70753154840549E-16
-1.16229834607148E-17
-2.32763511072116E-16
51
Table 4.3: Moment of Scaling Functions for N= 22
Moment of Scaling Functions for N= 22
1
2.31726465941445
5.36971550177098
12.2319705607159
26.8773186714991
55.6323219714068
104.846598989408
169.689702637117
205.772899131726
92.0630949558087
-326.991286254096
52
4.6 IMPACT LOAD
A discretized impulse time signal has shown in figure(4.5). The time window varies from
to The duration of time signal is and it starts at and ends at .
This load signal generated by using Gaussian function in the MATLAB.
Figure 4.5: Discredited impulse time signal
0 50 100 150 200 250 300 350 400 450 5000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (s)
F(t
)
54
CHAPTER 5
WAVELET BASED SPECTRAL FINITE ELEMENT FORMULATION
5.1 REDUCTION OF WAVE EQUATIONS TO ORDINARY DIFFERENTIAL
EQUATIONS
Governing wave equations for higher order composite beam [37] are,
where are the axial, transverses and shear displacements respectively.
where are the in-plane laminate maduli coefficients, in-plane/ flexure coupling
laminate moduli coefficients and flexural laminate stiffness coefficient. are the
inertial constants. These constants can be calculated by the following expression. is the mass
density , is the width of the beam and is the thickness of each ply of laminate. The force
boundary conditions associated with the governing differential equations are
55
where are the axial, transverse force and moment respectively. In
the formulation WSFE, reduce the governing differential equations (5.1)-(5.3) to ODE using
Daubechies scaling functions for approximation in time. The displacements
discredited at n points in the time window. Therefore the number of
sampling points the time at a particular instant can be obtained by
where is the time interval between two sampling points. Approximate the functions
and by scaling functions at an arbitrary scale as
Substitute above expressions in the governing wave equations we get
56
Taking inner product of on both sides of above equations with the translates of scaling functions
, and using their orthogonal properties, we get n simultaneous equations ordinary
differential equations are
where , is the order of the Daubechies wavelet and ,
are the
first and second order connection coefficients respectively.
For compactly supported wavelets these first and second order connection coefficients [8] are
non zeros at an interval of By observing the
equations(ODEs), the connection coefficients near the boundaries at lie
outside the time window [ ]. The falling of these connections coefficients outside the time
window will create problems with finite length data sequence. Therefore the coefficients at the
boundary need treatment. This treatment can be done using capacitance matrix and penalty
function methods [56,57]. To solve this boundary value problem, a wavelet based extrapolation
scheme proposed by Amaratunga and Williams(1997)[3,71] is used. This method is particularly
suitable for approximation time for the ease of imposition of initial conditions. The treatment of
57
boundaries for finite domain analysis can be done by using wavelet extrapolation technique. The
equation (5.11) can be written in matrix algebraic equations and can be solved by conventional
techniques. The equation (5.11) gives the algebraic coupled equations. are known in the
interval of time 0 to . Where varies from . Out of algebraic coupled equations,
some equations contain coefficients corresponding to to . This condition
shows that, the coefficients lie outside of the time window [0- ]. Similarly the same problem
exist on other boundary for to .
5.1.1 WAVELET EXTRAPOLATION TECHNIQUE
The coefficients of Daubechies scaling function has
vanishing moments, and that its
translates can be combined to give exact representation of polynomial order . Assume that
has a polynomial representation of order in the vicinity of the left boundary , we
have
where are constant coefficients. By taking inner product of equation (5.12) with on
both sides, we get
where are the moment of scaling functions defined in equation (4.17) and solving the
recursive equations in 4.4. Solve the equation (5.12) using finite difference scheme, we get
initial values of . Substitute these values in equation (5.13) and get the values in
58
terms of initial values. Like that the unknown coefficients on LHS can calculate by using the
following relation
1
1
0
1
2
1
2
0
2
1
2
1
2
0
2
1
1
1
1
0
1
2
2
1
pp
NNN
p
p
Nc
c
c
u
u
u
(521)
Similarly assume the polynomial representation to obtain the unknown coefficients at the
RHS as
The above equation (5.15) can be written in matrix form as
(5.23)
)1(
2)1(
1)1(
1
1
0
1
1
1
1
0
1
1
1
1
1
0
1
110
n
pn
pn
pp
p
ppp
p
ppp
u
u
u
c
c
c
calculate the values from the above equation (5.16), and substitute back in equation (5.15) for
, we get
(5.24)
1
1
0
1
2
1
2
0
2
1
1
1
1
0
1
1
0
1
0
0
0
21
1
pp
NNN
p
p
Nn
n
n
c
c
c
u
u
u
Finally, after getting the unknown coefficients on LHS and RHS, substitute these unknown
59
coefficients in equation (5.9), (5.10) and (5.11), and the system of coupled equations can be
written in matrix form as
where is the first order connection coefficient matrix obtain after using the wavelet
extrapolation technique. is the second order connection coefficient matrix[8] evaluated
independently, and this can also be written as . This modification of second order
connection coefficient matrix will helps in the imposition of initial conditions for non periodic
solutions[43]. The reduced ordinary differential equations (ODE) are coupled in wavelet spectral
finite element (WSFE). Decoupling of these equations can be done by eigenvalue analysis of
connection coefficient matrix and can be written as
Here is the eigen vector matrix of and is the diagonal matrix containing corresponding
eigenvalues . Similarly the eigenvalue analysis of second order connection coefficient
can be written as
Here, is a diagonal matrix with diagonal terms . These matrices and are
independents of the problem and depends only on the order of wavelet . After decoupling of
the equations (5.18),(5.19) and (5.20) can be written as
60
where and . Similarly the force boundary conditions
given by (5.6)-(5.8) can be written as
where and and are the transformed force boundary conditions of
and respectively.
5.1.2 CALCULATION OF WAVE NUMBERS AND WAVE AMPLITUDES
The parameters such as wave numbers and wave speeds are required to understand the wave
mechanics in the wave guide and these are required in SFE formulation to know the wave
characteristics like the wave mode is in propagating mode or damping mode or combination of
both. This spectral analysis starts with the partial differential equations (5.1)-(5.3) governing the
waveguide. and are the field variables in spatial and temporal
dimensions. These field variables are transformed to frequency domain by using DFT as
61
Here is the discrete circular frequency in , is the total number of frequency points
used in the transformation and , and are the DFT coefficients, varies with only. The
discrete circular frequency related with the time window as
where is the time sampling rate and is the highest frequency captured by decides the
frequency content of the load and considering the wrap around problem and decides the
aliasing problem [18]. Substitute the equations (5.36)-(5.38) in equations (5.1)-(5.3), we get
Assuming the solutions of above equations as ,
and .
Here and are the constants, later these can be derived from boundary values and k is the
wave number. Substituting these solutions in equations (5.39)-(5.41), we get
The above equations can also be write in matrix form, we get
62
These wave numbers k were obtained as a function of circular frequency , and computation of
wave numbers are very difficult for higher composite structures due to its coupling of transverse
and shear displacements and elastic coupling. In order to calculate the wave numbers and
associated wave amplitudes for such problems have been proposed by S. Gopalakrishnan [21].
Singular value decomposition (SVD) and polynomial evaluation problem (PEP) methods are
used to solve these problems. In this project, PEP method is used to solve the equation (5.42)-
(5.44) and those equations are written in the form PEP in k as
Here are the matrices, where is the number of independent variables in
the governing equations. Thus, the PEP is of the order . In this project, and .
The PEP for equation(5.45) is given as
where
By solving the equation (5.47)`using “polyeig” in the MATLAB, gives the eigenvalues and
eigenvectors. These eigenvalues represents the wavenumbers and eigenvectors represents the
corresponding wave amplitudes.
5.2 SPECTRAL FINITE ELEMENT FORMULATION
The equations (5.39) to (5.41) are solved exactly to get the shape functions in terms of some
unknown constants. These unknown constants are again solved in terms of boundary values and
these are useful in the formulation of elemental dynamic stiffness matrix of the transformed
nodal displacements with transformed nodal forces.
63
Figure 5.1: Composite beam element with nodal forces and nodal displacements
The above figure shows the beam element with three degrees of freedom such as , and
represents the axial, transverse and shear displacements at node1 and node2 respectively and
, and are the axial, transverse forces and moments at node1 and node2 respectively.
The transformed ODEs (5.30) to (5.32) are need to be solved for the , and and the actual
solutions , and are obtained from inverse wavelet transform and solve
these equations exactly, gives the exact shape functions (interpolating functions) which are
where L is the length of the element and and are the wave numbers corresponding to
the axial, transverse and shear modes of the element and explained in Mitra & Gopalakrishnan
(in press)
64
The above equations (5.48)-(5.50) can be written in matrix form as
161514131211 RRRRRR
262524232221 RRRRRR
363534333231 RRRRRR
)(ik-
ik-
)(ik-
ik-
)(ik-
3
3
2
2
1
1
e
e
e
e
e
xL
x
xL
x
xL
xike
6
5
4
3
2
1
c
c
c
c
c
c
Here is the nodal displacement vector of element at each node. [ ] is the amplitude
ratio matrix and this matrix can be obtained by solving the PEP equation (5.47). The solution of
equation (5.47) gives the eigenvalues and eigenvectors. [ ] is the eigenvector matrix and
eigenvalues are the wave numbers and . is the diagonal matrix with the
diagonal elements xike 1 )(-ik1e
xL x2-ike )(-ik2e
xL x3-ike )(-ik3e
xL and
are the unknown coefficients and these are obtained by applying the
boundary conditions at node 1 and node 2 as shown in figure 5.1.
Applying the boundary conditions at two nodes of element as shown in figure 5.1 to get
unknown constants in terms of nodal displacements. Put in equation (5.51), gives the
nodal displacements at node1, and the equation becomes
Put in equation (5.51), gives the nodal displacements at node2, and the equation becomes
65
Here
is the nodal displacement vector at node 1, similarly
is the nodal displacement vector at node 2. and are the diagonal matrices
and these are shown in the next page.
1
e
1
e
1
e
e
1
e
1
e
1
3
2
1
3
2
1
ik-
ik-
-ik
2
ik-
ik-
ik-
1
L
L
L
L
L
L
From equations (5.52) and (5.53), the nodal displacement vector
of an element
can be written as
where
and and similarly
.
The derivatives of the interpolating functions are given as
66
The above equations (5.55)-(5.57) can be written in matrix form as
6
5
4
3
2
1
)(ik-
ik-
)(ik-
ik-
)(ik-
363353342332321311
263253242232221211
163153142132121111
3
3
2
2
1
1
e
e
e
e
e
=
c
c
c
c
c
ce
RikRikRikRikRikRik
RikRikRikRikRikRik
RikRikRikRikRikRik
M
V
P
xL
x
xL
x
xL
xik
Here is the nodal force vector of element at each node. is the diagonal matrix with the
diagonal elements xike 1 )(-ik1e
xL x2-ike )(-ik2e
xL x3-ike )(-ik3e
xL and
are the unknown coefficients and these are obtained by applying the
boundary conditions at node 1 and node 2 as shown in figure 5.1.
Applying the boundary conditions at two nodes of element as shown in figure 5.1 to get
unknown constants in terms of nodal forces. Put in equation (5.58), gives the nodal
forces at node1, and the equation becomes
Put in equation (5.58), gives the nodal displacements at node2, and the equation becomes
Here
is the nodal force vector at node 1, similarly
is the
nodal force vector at node 2. and are the diagonal matrices as given above.
From equations (5.59) and (5.60), the nodal force vector
of an element can be
written as
67
where
and and similarly
.
From equation (5.54),
Substitute the equation (5.62) in equation (5.61) we get
The above equation can also be written as
where is the elemental dynamic stiffness matrix of size and this matrix is divided
into four sub matrices of size , and rewrite the equation (5.64) as
where and
are the nodal force and displacement vectors of the element at node.
5.3 MODELING OF DE-LAMINATION IN COMPOSITE BEAM
The modeling of embedded de-lamination of the composite beam is done according to the
method presented by Nag et al.[51]. The following figure 5.2 shows the embedded de-lamination
of the graphite – epoxy cantilevered beam.
68
where L is the length of the beam, is the distance between the free end and edge of the de-
lamination, is the de-lamination length, 2h is the depth of the beam and b is the width of the
beam. Figure 5.3(a) and 5.3(b) are shows the cross sections of the beam at the end and at the de-
laminated portion of the composite beam
Figure 5.3(a): Cross section at the end Figure 5.3(b): Cross section at the de-lamination
Figure 5.4: Modeling of an embedded de-lamination with base and sub laminates.
Figure 5.5: Representation of the base and sub base laminates by spectral elements.
Figure 5.2 shows the delaminated composite beam divided into four elements as shown in
Figure5.4, out of four elements element 1 and element 2 are called as base laminates and element
3 and element 4 are called as sub laminates. Every element of the beam is considered as
individual structural wave guides and modeled as coupled composite Timoshenko beam by using
WSFE method as discussed in previous section.
From equation (5.65), the nodal force vector for element 1is written as
69
Similarly, the nodal force vector for element 2 is written as
The kinematic assumption at the interfaces of the base and sub laminates is that the constant
cross sectional slope. From this assumption, the constant and continuous slope at the two
interfaces between base laminates and sub base laminates gives relation of nodal displacement
vector of sub laminates in term of base laminates.
Figure 5.6: Force balance at the interface between base and sub laminate elements.
From the above Figure, the nodal displacement vector at node 3 and node 5 is written in terms of
node 4 as
From the above expression and
can be written as
70
Similarly, the nodal displacement vector at node 6 and node 8 is written in terms of node 7 as
From the above expression and
can be written as
where represents the nodal displacement vector of element at node. and are the
transformation matrices in terms of top and bottom sub laminate thicknesses and .
From the force equilibrium equation at interface AB as shown in figure (5.5)
The equation (5.70) can be written as
Similarly, the force equilibrium equation at interface CD as shown in figure (5.5)
The equation (5.72) can be written as
The nodal force vector for element 3 is written as
71
Substitute and
values in equation (5.74) and pre-multiplying with on both sides, we get
Similarly, The nodal force vector for element 4 is written as
Substitute and
values in equation (5.76) and pre-multiplying with on both sides, we get
After calculation of elemental nodal force vectors, assemble the four spectral elements, we get
Nodal displacements at fixed end are zero i.e. at node 1, and nodal forces vectors are zero
at node 4 and node 7. Using these conditions, the equation (5.78) can be written as
where is the reconstructed dynamic stiffness matrix for the spectral element with embedded
de-lamination of cantilever composite beam.
73
CHAPTER 6
NUMERICAL EXAMPLES AND CONCLUSIONS
The wave propagation of the composite beam is obtained by using WSFE as discussed in the
previous chapter for broad band impulse load as shown in Figure (4.5). These numerical
experiments are done over an eight layered AS4/3501-6 graphite-epoxy composite beam. The
properties of AS4/3501-6 graphite-epoxy composite beam are given in the following table 6.1.
Table 6.1: Material properties of the Graphite- Epoxy composite beam:
Material properties
141.90 Gpa.
9.78 Gpa.
6.13 Gpa.
4.80 Gpa
0.42
Mass density ( ) 1449 kg/m3
The length (L), breadth (2b) and depth (2h) of the beam are 0.5m, 0.01m and 0.01m respectively
as shown in Figure (5.2). The de-lamination is provided at a distance of 0.25m ( ) from free end
of the cantilever beam and extends towards the fixed end at mid depth and at various depths
above the center line of the beam.
74
6.1 WAVE RESPONSES TO THE IMPULSE LOAD
Wave propagation is studied in a AS4/3501-6 graphite-epoxy cantilevered composite beam due
to the impulse load applied axially and transversely at the free end. The Figure (6.1) and (6.2),
shows the axial and transverse velocities in a undamped [ ] cantilever composite beam
respectively. These velocities are simulated using WSFE with , with total time
window and for the length, width and depth of beam are 0.5m, 0.01m and 0.01m
respectively.
Figure 6.1: Axial tip velocity of undamaged [ ] composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
X: 0.000324
Y: 0.001386
Time (s)
Axia
l V
elo
city (
m/s
)
75
Figure 6.2: Transverse tip velocity of undamaged [ ] composite beam
The figure (6.3) and (6.4) shows the axial and transverse tip velocities of the beam of same
configuration as mentioned in the above plot, but here different length of de-lamination ( ) is
embedded in the beam and compared with undamaged response. The de-laminations are
embedded along the centerline of the beam and the de-lamination starts at a distance of 0.25m
from free end and extends towards fixed end with different length of 10mm and 20mm. From
figure (6.3), it is clearly observed that, the axial velocity of undamaged beam and de-lamination
of 10mm and 20mm produced the same response. In case of transverse velocity, the damaged
responses compared with undamaged response shows the early reflections from the fixed end
due to its de-lamination and amplitude of these reflected waves increases with increase in the
length of de-lamination which can be observed from figure (6.4). Figures (6.5) and (6.6) shows
the axial and transverse tip velocities for undamaged and different length of de-lamination
(30mm and 40 mm) along the centerline of the eight layered composite beam.
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-4
-2
0
2
4
6
8
10x 10
-3
X: 0.000128
Y: 0.008161
Time (s)
Tra
nsvers
e V
elo
city (
m/s
)undamaged
76
Figure 6.3: Axial tip velocities for undamaged and different de-lamination lengths ( =10mm
and 20mm) along the centerline of [ ] layered composite beam
Figure 6.4: Transverse tip velocities for undamaged and different de-lamination lengths
( =10mm and 20mm) along the centerline of [ ] composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Axia
l V
elo
city (
m/s
)
undamaged
10 mm delamination
20 mm delamination
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-4
-2
0
2
4
6
8
10x 10
-3
Time (s)
Tra
nsvers
e V
elo
city (
m/s
)
undamaged
10 mm delamination
20 mm delamination
77
Figure 6.5: Axial tip velocities for undamaged and different de-lamination lengths ( =30mm
and 40mm) along the centerline of [ ] layered composite beam
Figure 6.6: Transverse tip velocities for undamaged and different de-lamination lengths
( =30mm and 40mm) along the centerline of [ ] composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Axia
l V
elo
city (
m/s
)
undamaged
30 mm delamination
40 mm delamination
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-4
-2
0
2
4
6
8
10x 10
-3
Time (s)
Tra
nsvers
e V
elo
city (
m/s
)
undamaged
30 mm delamination
40 mm delamination
78
Figure 6.7: Axial tip velocities for undamaged and different position (0.25m and 0.35m) of
20mm de-lamination from free end and along the centerline of [ ] layered composite beam
Figure 6.8: Transverse tip velocities for undamaged and different position (0.25m and 0.35m) of
20mm de-lamination from free end and along the centerline of [ ] composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Axia
l V
elo
city (
m/s
)
undamaged
20mm delamination @ 0.25m
20mm delamination @ 0.35m
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-4
-2
0
2
4
6
8
10x 10
-3
Time (s)
Tra
nsvers
e V
elo
city (
m/s
)
undamaged
20mm delamination @ 0.25m
20mm delamination @ 0.35m
79
The figure (6.7) and (6.8) shows the axial and transverse tip velocities of undamaged beam and
20mm de-lamination at different positions from the free end and along the centerline of the [ ]
layered composite beam. From figure (6.7), it is clearly observed that, the axial velocity of
undamaged beam and 20mm de-lamination at different positions from the free end and along the
centerline of the composite beam produce the same response. From figure (6.8), it can be
observed that the position of reflections due to the de-lamination changes by changing the de-
lamination position.
Figure 6.9: Axial tip velocities for undamaged and different depths above the centerline (h1=h,
h1=h/2 and h1=h/4) of 20mm de-lamination of [ ] layered composite beam
From figure (6.9) it can be observed that the response of undamaged and 20mm de-lamination at
=h i.e. at the centerline of the beam is same in the axial tip velocity. The response, for h1=h/2
and h1=h/4 gives undulations in the axial tip velocity. Where h1 is the depth of de-lamination
from the top surface of the beam.
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-1.5
-1
-0.5
0
0.5
1
1.5x 10
-3
Time (s)
Axia
l V
elo
city (
m/s
)
undamaged
h1=h
h1=h/2
h1=h/4
80
Figure 6.10: Transverse tip velocities for undamaged and different depths above the centerline
(h1=h, h1=h/2 and h1=h/4) of 20mm de-lamination of [ ] layered composite beam
Figure 6.11: Axial tip velocities for different orientation ( , , , and ) of
20mm de-lamination along the centerline of composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-4
-2
0
2
4
6
8
10x 10
-3
Time (s)
Tra
nsvers
e V
elo
city (
m/s
)
undamaged
h1=h
h1=h/2
h1=h/4
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-5
-4
-3
-2
-1
0
1
2
3
4
5x 10
-3
Time (s)
Axia
l V
elo
city (
m/s
)
[0] 8layers
[30] 8layers
[45] 8layers
[60] 8layers
81
Figure (6.10) shows the transverse tip velocity of undamaged composite beam compared
with different depths above the centerline ( =h, =h/2 and =h/4) of 20mm de-lamination of
beam. Undulations can be observed from figure (6.10) by reducing the by which number of
undulations increases and the amplitude of these undulations initially decreases and later on
increases. In figure (6.11) and (6.12), comparing the axial and transverse tip velocities for
different ply orientation of , , , and layered composite beam with 20mm
de-lamination along the centerline. From these figures, it can be observed that the amplitude of
the undulations increases and shifting of these undulations occurs by increasing the ply
orientation.
Figure 6.12: Transverse tip velocities for different orientation ( , , , and ) of
20mm de-lamination along the centerline of composite beam
0 0.2 0.4 0.6 0.8 1 1.2
x 10-3
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Time (s)
Tra
nsve
rse
Vel
ocity
(m
/s)
[0] 8layers
[30] 8layers
[45] 8layers
[60] 8layers
83
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